Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).
However, not all Coxeter groups are finite, and not all can be described in terms of symmetries and Euclidean reflections.
[2] Coxeter groups find applications in many areas of mathematics.
is stipulated in the definition is that together with already implies that An alternative proof of this implication is the observation that
, but the elements are modified, being proportional to the dot product of the pairwise generators.
The indefinite type is sometimes further subdivided, e.g. into hyperbolic and other Coxeter groups.
However, there are multiple non-equivalent definitions for hyperbolic Coxeter groups.
are placed in a row with each vertex joined by an unlabelled edge to its immediate neighbors is the Coxeter diagram of the symmetric group
On the other hand, reflection groups are concrete, in the sense that each of its elements is the composite of finitely many geometric reflections about linear hyperplanes in some euclidean space.
Each generator of a Coxeter group has order 2, which abstracts the geometric fact that performing a reflection twice is the identity.
, corresponding to the geometric fact that, given two hyperplanes meeting at an angle of
[1] The converse is partially true: every finite Coxeter group admits a faithful representation as a finite reflection group of some Euclidean space.
that are not exceptionally isomorphic to a Weyl group (namely
[6] Dynkin diagrams have the additional restriction that the only permitted edge labels are 2, 3, 4, and 6, which yields the above.
Geometrically, this corresponds to the crystallographic restriction theorem, and the fact that excluded polytopes do not fill space or tile the plane – for
Note further that the (directed) Dynkin diagrams Bn and Cn give rise to the same Weyl group (hence Coxeter group), because they differ as directed graphs, but agree as undirected graphs – direction matters for root systems but not for the Weyl group; this corresponds to the hypercube and cross-polytope being different regular polytopes but having the same symmetry group.
Some properties of the finite irreducible Coxeter groups are given in the following table.
The exceptional regular polytopes in dimensions two, three, and four, correspond to other Coxeter groups.
For n = 2, this can be pictured as a subgroup of the symmetry group of the standard tiling of the plane by equilateral triangles.
In general, given a root system, one can construct the associated Stiefel diagram, consisting of the hyperplanes orthogonal to the roots along with certain translates of these hyperplanes.
[9] The Stiefel diagram divides the plane into infinitely many connected components called alcoves, and the affine Coxeter group acts freely and transitively on the alcoves, just as the ordinary Weyl group acts freely and transitively on the Weyl chambers.
Then the affine Coxeter group is generated by the ordinary (linear) reflections about the hyperplanes perpendicular to
The Coxeter graph for the affine Weyl group is the Coxeter–Dynkin diagram for
In this case, one alcove of the Stiefel diagram may be obtained by taking the fundamental Weyl chamber and cutting it by a translate of the hyperplane perpendicular to
A choice of reflection generators gives rise to a length function ℓ on a Coxeter group, namely the minimum number of uses of generators required to express a group element; this is precisely the length in the word metric in the Cayley graph.
An element v exceeds an element u in the Bruhat order if some (or equivalently, any) reduced word for v contains a reduced word for u as a substring, where some letters (in any position) are dropped.
The Hasse diagrams corresponding to these orders are objects of study, and are related to the Cayley graph determined by the generators.
is generated by finitely many elements of order 2, its abelianization is an elementary abelian 2-group, i.e., it is isomorphic to the direct sum of several copies of the cyclic group
, was computed in (Ihara & Yokonuma 1965) for finite reflection groups and in (Yokonuma 1965) for affine reflection groups, with a more unified account given in (Howlett 1988).
of finite or affine Weyl groups, the rank of
